Question. Suppose $d$ is even. How many connected components $\mathcal U_{n,d}$ has
? Do we know anything about
the Betti numbers of $\mathcal U_{n,d}$ ?

Indeed, motivated by this other question, I am trying to figure out if it makes sense
to ask for the number of connected components of the space of polynomial contact
distributions of even degree on $\mathbb P^{2n+1}_{\mathbb R}$.

1 Answer
1

If $s$ is a non-zero section whose image lies in $\mathcal U_{n,d}$, then it has constant sign on $V^\ast:=\mathbb R^{n+1}\setminus\{0\}$ and after possibly multiplying by $-1$ we may assume that $s$ is strictly positive on $V^\ast$. The strictly positive $s$ form an open convex cone $C$ (we do not assume that $0$ belongs to a cone) and is hence contractible when non-empty which this one is when $d$ is even. As $C\to\mathcal U_{n,d}$ is a fibration with fibres $\mathbb R_+$ so is $\mathcal U_{n,d}$.